Andrews plots provide aesthetically pleasant visualizations of high-dimensional datasets. This work proves that Andrews plots (when defined in terms of the principal component scores of a dataset) are optimally ``smooth'' on average, and solve an infinite-dimensional quadratic minimization program over the set of linear isometries from the Euclidean data space to $L^2([0,1])$. By building technical machinery that characterizes the solutions to general infinite-dimensional quadratic minimization programs over linear isometries, we further show that the solution set is (in the generic case) a manifold. To avoid the ambiguities presented by this manifold of solutions, we add ``spectral smoothing'' terms to the infinite-dimensional optimization program to induce Andrews plots with optimal spatial-spectral smoothing. We characterize the (generic) set of solutions to this program and prove that the resulting plots admit efficient numerical approximations. These spatial-spectral smooth Andrews plots tend to avoid some ``visual clutter'' that arises due to the oscillation of trigonometric polynomials.

翻译：Andrews图提供了高维数据集的视觉愉悦可视化。本研究证明，当Andrews图基于数据集的主成分得分定义时，其在平均意义上是最优“平滑”的，并且它求解了从欧几里得数据空间到L^2([0,1])的线性等距集合上的无限维二次极小化问题。通过构建刻画线性等距上一般无限维二次极小化问题解的技术机制，我们进一步证明解集（在一般情形下）是一个流形。为避免该解流形带来的歧义，我们在无限维优化问题中引入“谱平滑”项，以生成具有最优空间-谱平滑的Andrews图。我们刻画了该问题（一般情形下的）解集，并证明所得图形可实现高效数值逼近。这些空间-谱平滑Andrews图能够有效避免因三角多项式振荡而产生的“视觉杂乱”现象。